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# stbsv

```
NAME
STBSV - solve one of the systems of equations   A*x = b, or
A'*x = b,

SYNOPSIS
SUBROUTINE STBSV ( UPLO, TRANS, DIAG, N, K, A, LDA, X, INCX
)

INTEGER      INCX, K, LDA, N

CHARACTER*1  DIAG, TRANS, UPLO

REAL         A( LDA, * ), X( * )

PURPOSE
STBSV  solves one of the systems of equations

where b and x are n element vectors and A is an n by n unit,
or non-unit, upper or lower triangular band matrix, with ( k
+ 1 ) diagonals.

No test for singularity or near-singularity is included in
this routine. Such tests must be performed before calling
this routine.

PARAMETERS
UPLO   - CHARACTER*1.
On entry, UPLO specifies whether the matrix is an
upper or lower triangular matrix as follows:

UPLO = 'U' or 'u'   A is an upper triangular matrix.

UPLO = 'L' or 'l'   A is a lower triangular matrix.

Unchanged on exit.

TRANS  - CHARACTER*1.
On entry, TRANS specifies the equations to be solved
as follows:

TRANS = 'N' or 'n'   A*x = b.

TRANS = 'T' or 't'   A'*x = b.

TRANS = 'C' or 'c'   A'*x = b.

Unchanged on exit.

DIAG   - CHARACTER*1.
On entry, DIAG specifies whether or not A is unit
triangular as follows:

DIAG = 'U' or 'u'   A is assumed to be unit triangu-
lar.

DIAG = 'N' or 'n'   A is not assumed to be unit tri-
angular.

Unchanged on exit.

N      - INTEGER.
On entry, N specifies the order of the matrix A.  N
must be at least zero.  Unchanged on exit.

K      - INTEGER.
On entry with UPLO = 'U' or 'u', K specifies the
number of super-diagonals of the matrix A.  On entry
with UPLO = 'L' or 'l', K specifies the number of
sub-diagonals of the matrix A.  K must satisfy  0
.le. K.  Unchanged on exit.

A      - REAL             array of DIMENSION ( LDA, n ).
Before entry with UPLO = 'U' or 'u', the leading ( k
+ 1 ) by n part of the array A must contain the upper
triangular band part of the matrix of coefficients,
supplied column by column, with the leading diagonal
of the matrix in row ( k + 1 ) of the array, the
first super-diagonal starting at position 2 in row k,
and so on. The top left k by k triangle of the array
A is not referenced.  The following program segment
will transfer an upper triangular band matrix from
conventional full matrix storage to band storage:

DO 20, J = 1, N M = K + 1 - J DO 10, I = MAX( 1, J -
K ), J A( M + I, J ) = matrix( I, J ) 10    CONTINUE
20 CONTINUE

Before entry with UPLO = 'L' or 'l', the leading ( k
+ 1 ) by n part of the array A must contain the lower
triangular band part of the matrix of coefficients,
supplied column by column, with the leading diagonal
of the matrix in row 1 of the array, the first sub-
diagonal starting at position 1 in row 2, and so on.
The bottom right k by k triangle of the array A is
not referenced.  The following program segment will
transfer a lower triangular band matrix from conven-
tional full matrix storage to band storage:

DO 20, J = 1, N M = 1 - J DO 10, I = J, MIN( N, J + K
) A( M + I, J ) = matrix( I, J ) 10    CONTINUE 20
CONTINUE

Note that when DIAG = 'U' or 'u' the elements of the
array A corresponding to the diagonal elements of the

matrix are not referenced, but are assumed to be
unity.  Unchanged on exit.

LDA    - INTEGER.
On entry, LDA specifies the first dimension of A as
declared in the calling (sub) program. LDA must be at
least ( k + 1 ).  Unchanged on exit.

X      - REAL             array of dimension at least
( 1 + ( n - 1 )*abs( INCX ) ).  Before entry, the
incremented array X must contain the n element
right-hand side vector b. On exit, X is overwritten
with the solution vector x.

INCX   - INTEGER.
On entry, INCX specifies the increment for the ele-
ments of X. INCX must not be zero.  Unchanged on
exit.

Level 2 Blas routine.

-- Written on 22-October-1986.  Jack Dongarra,
Argonne National Lab.  Jeremy Du Croz, Nag Central
Office.  Sven Hammarling, Nag Central Office.
Richard Hanson, Sandia National Labs.
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